Jürgen Neukirch

Pity the poor bibliographer. This book by Jürgen Neukirch entitled Class Field Theory is not the same as the book by Jürgen Neukirch entitled Class Field Theory which was published by Springer in 1986 as volume 280 in their Grundlehren der mathematischen Wissenschaften series. This one was published by Springer in 2013 in their Universitext series. But this one is the earlier book, originally published in German in 1969 (it appears as reference 42 in the Grundlehren volume). And that was a reprint of an earlier edition that had appeared in the Bonner Mathematischen Schriften series and based on lectures Neukirch had given in 1965–66. Alexander Schmidt tells us he has “corrected mistakes and updated notation” in the German text, which has now been translated by F. Lemmermeyer and W. Snyder. (The books also has a “language editor”, A. Rosenschon, but I’m not sure what such a person does.)

In the introduction to his 1986 book, Neukirch says

Class field theory, which is so immediately compelling in its main assertions, has, ever since its invention, suffered from the fact that its proofs have required a complicated and, by comparison with the results, rather imperspicuous system of arguments which have tended to jump around all over the place. My earlier presentation of the theory [41] (sic!) has strengthened me in my belief that a highly elaborate mechanism, such as, for example, cohomology, might not be adequate for a number0theoretical law admitting a very direct formulation, and that the truth of such a law must be susceptible to a far more immediate insight.

Indeed, I have noted this difficulty myself in my review of another book called Class Field Theory. In 1986, Neukirch thought he had found a better way and hence wrote a new book. There is also a treatment of class field theory in Neukirch’s Algebraic Number Theory, which I have not read.

It seems, however, that Neukirch’s assessment of his older notes did not convince others, because Schmidt tells us that

Although this book has been out of print for many years, it has remained the favorite introduction to class field theory in Germany. As a student in the 1980s, I myself studied a copy from the library that showed clear signs of extensive use. This motivated the idea to make the text available again, as a printed book as well as a freely accessible file for downloading.

The old-new Class Field Theory — The Bonn Lectures uses the approach to class field theory via cohomology of groups. It is done cleanly and well, as one would expect from Neukirch, and so it would certainly serve as a place to begin learning the theory. One very useful section appears at the very end, when Neukirch translates the cohomological language in which he has expressed the results to obtain the classical ideal-theoretic formulation.

Neukirch’s point from his 1986 book stands, however: I’m not aware of a fully satisfactory account of class field theory. This one is as good as they come, but perhaps we still don’t understand it well enough to write a good book about it.

Fernando Q. Gouvêa will probably never get the chance to teach a course in class field theory, so he may never really understand it.

Part II: Local Class Field Theory
1. Abstract Class Field Theory
2. Galois Cohomology
3. The Multiplicative Group of a \(\mathfrak{p}\)-adic Number Field
4. The Class Formation of Unramified Extensions
5. The Local Reciprocity Law
6. The Existence Theorem
7. Explicit Determination of the Norm Residue Symbol

Part III: Global Class Field Theory
1. Number Theoretic Preliminaries
2. Idèles and Idèle Classes
3. Cohomology of the Idèle Group
4. Cohomology of the Idèle Class Group
5. Idèle Invariants
6. The Reciprocity Law
7. The Existence Theorem
8. The Decomposition Law
9. The Ideal Theoretic Formulation of Class Field Theory